Mathematical Biosciences and Engineering, 2013, 10(4): 1067-1094. doi: 10.3934/mbe.2013.10.1067.

Primary: 49J15, 49N90; Secondary: 93C10, 93C95.

Export file:

Format

  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text

Content

  • Citation Only
  • Citation and Abstract

Parametrization of the attainable set for a nonlinear control model of a biochemical process

1. Department of Mathematics and Computer Sciences, Texas Woman's University, Denton, TX 76204
2. Department of Computer Mathematics and Cybernetics, Moscow State Lomonosov University, Moscow, 119992
3. Centre de Recerca Matemàtica, Campus de Bellaterra, Edifici C, 08193 Bellaterra, Barcelona

   

In this paper, we study a three-dimensional nonlinear model of a controllable reaction$ [X] + [Y] + [Z] \rightarrow [Z] $, where the reaction rate is given by a unspecifiednonlinear function. A model of this type describes a variety of real-life processes in chemicalkinetics and biology; in this paper our particular interests is in its application to wastewater biotreatment. For this control model, we analytically study the corresponding attainableset and parameterize it by the moments of switching of piecewise constant control functions.This allows us to visualize the attainable sets using a numerical procedure.
    These analytical results generalize the earlier findings, which were obtained for a trilinearreaction rate (which corresponds to the law of mass action) and reportedin [18,19], to the case of a general rate of reaction. These results allow toreduce the problem of constructing the optimal control to a straightforward constrained finitedimensional optimization problem.
  Figure/Table
  Supplementary
  Article Metrics

Keywords waste water biotreatment; Environmental problem; nonlinear reaction rate; attainable set.

Citation: Ellina Grigorieva, Evgenii Khailov, Andrei Korobeinikov. Parametrization of the attainable set for a nonlinear control model of a biochemical process. Mathematical Biosciences and Engineering, 2013, 10(4): 1067-1094. doi: 10.3934/mbe.2013.10.1067

References

  • 1. Springer, Berlin-New York, 1984.
  • 2. Computers & Chemical Engineering, 34 (2010), 802-811.
  • 3. Springer-Verlag, Berlin-Heidelberg-New York, 2003.
  • 4. Sarsia, 85 (2000), 211-225.
  • 5. Environmental Technology Letters, 6 (1985), 467-476.
  • 6. Mathematics-in-Industry Case Studies Journal, 2 (2010), 34-63.
  • 7. Bulletin of Mathematical Biology, 52 (1990), 677-696.
  • 8. Journal of Mathematical Sciences, 12 (1979), 310-353.
  • 9. CRS Press, Boca Raton, Florida, 1994.
  • 10. Nauka, Moscow, 1967.
  • 11. SIAM Journal of Control and Optimization, 30 (1992), 1087-1091.
  • 12. SIAM Journal on Applied Mathematics, 67 (2007), 337-353.
  • 13. Computer Aided Chemical Engineering, 28 (2010), 967-972.
  • 14. Moscow University. Computational Mathematics and Cybernetics, (2001), 27-32.
  • 15. Dynamical Systems and Differential Equations, (2003), 359-364.
  • 16. Moscow University. Computational Mathematics and Cybernetics, (2005), 23-28.
  • 17. Journal of Dynamical and Control Systems, 11 (2005), 157-176.
  • 18. Neural, Parallel, and Scientific Computations, 20 (2012), 23-35.
  • 19. in "Industrial Waste" (eds. K.-Y. Show and X. Guo), InTech, Croatia, (2012), 91-120.
  • 20. Journal of Theoretical Biology, 227 (2004), 349-358.
  • 21. Differential Equations, 45 (2009), 1636-1644.
  • 22. Journal of Applied Mathematics and Mechanics, 62 (1998), 169-175.
  • 23. Jorn Wiley & Sons, New York-London-Sydney, 1964.
  • 24. Giorn. Ist. Ital. Attuari, 7 (1936), 74-80.
  • 25. Mathematical Notes, 37 (1985), 916-925.
  • 26. Mathematical Medicine and Biology, 26 (2009), 309-321.
  • 27. Mathematical Medicine and Biology, 26 (2009), 225-239.
  • 28. Discrete and Continuous Dynamical System. Ser. B, 14 (2010), 1095-1103.
  • 29. American Mathematical Society, Providence, RI, 1968.
  • 30. Birkhäuser, Boston, 1997.
  • 31. Mathematical Medicine and Biology, 27 (2010), 157-179.
  • 32. Mathematical Biosciences and Engineering, 8 (2011), 307-323.
  • 33. Journal of Mathematical Biology, 64 (2012), 557-577.
  • 34. Jorn Wiley & Sons, New York, 1967.
  • 35. Mat. Zametki, 41 (1987), 71-76, 121.
  • 36. Control and Cybernetics, 36 (2007), 5-45.
  • 37. Journal of Applied Mathematics and Mechanics, 46 (1982), 590-595.
  • 38. Mathematical Notes, 27 (1980), 429-437, 494.
  • 39. Journal of Optimization Theory and Applications, 64 (1990), 349-366.
  • 40. Journal of Optimization Theory and Applications, 64 (1990), 367-377.
  • 41. Springer-Verlag, Berlin-Heidelberg-New York, 1983.
  • 42. Bulletin of the American Mathematical Society, 48 (1942), 883-890.
  • 43. Springer, New York-Heidelberg-Dordrecht-London, 2012.
  • 44. Hermann, Paris, 1967.
  • 45. Computational Mathematics and Mathematical Physics, 47 (2007), 1768-1778.
  • 46. Computational Mathematics and Mathematical Physics, 51 (2011), 537-549.
  • 47. IEEE Transactions on Curcuits and Systems, 32 (1985), 503-505.
  • 48. Discrete and Continuous Dynamical Systems. Ser. B, 3 (2003), 361-382.
  • 49. SIAM Journal on Applied Mathematics, 61 (2000), 803-833.
  • 50. SIAM Journal on Applied Mathematics, 61 (2000), 983-1012.
  • 51. Springer-Verlag, Berlin-Heidelberg-New York, 1985.
  • 52. Automation and Remote Control, 70 (2009), 772-786.
  • 53. Automation and Remote Control, 72 (2011), 1291-1300.
  • 54. Journal of Mathematical Sciences, 139 (2006), 6863-6901.
  • 55. AMS, 2008.

 

This article has been cited by

  • 1. E.V. Grigorieva, E.N. Khailov, S. Anita, N. Hritonenko, G. Marinoschi, A. Swierniak, Optimal Vaccination, Treatment, and Preventive Campaigns in Regard to the SIR Epidemic Model, Mathematical Modelling of Natural Phenomena, 2014, 9, 4, 105, 10.1051/mmnp/20149407
  • 2. E.V. Grigorieva, E.N. Khailov, A. Korobeinikov, A. Morozov, S. Petrovskii, Optimal Control for a SIR Epidemic Model with Nonlinear Incidence Rate, Mathematical Modelling of Natural Phenomena, 2016, 11, 4, 89, 10.1051/mmnp/201611407

Reader Comments

your name: *   your email: *  

Copyright Info: 2013, , licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved